If you are flying long haul into Japan, it'd be better to take it easy your first day and not try and paint the town red in Kanazawa. If you have the energy, you can see some place in Tokyo, since it never sleeps, otherwise take a nice hot bath and throw back a few drinks to relax. Or your second idea of putting off Kanazawa for a future trip works well too - as said, there is indeed a lot in the area.
Two days to see Hiroshima City and Miyajima are fine. Get an early start to see Miyajima, and if you can, see the shrine at high tide and walk out on the beach to touch the big torii at low tide.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-iCNZ6AdsKGUB-HPKNxP1...Don't miss the Daishoin Temple - it is in many ways one of the very best sights on the island. You actually have the option to see many other great places too. Iwakuni is just a stone's throw from Miyajima, and the iconic bridge and mountaintop castle are very enjoyable.
https://www.youtube.com/watch?v=0COOZXi66mMAnother good place is Onomichi, with its temple tour and even better is the Kosanji Temple which is simply mind blowing in its color and detail.
https://www.youtube.com/watch?v=97Q5EjEOXI8Takamatsu is really one of the most interesting cities on Shikoku. While the Sanuki Udon may be famous, the real treat is the Ritsurin Garden, and you should not miss it.
https://www.youtube.com/watch?v=afx-Ju52FiYAnd if you're going all the way there, along the way is Kotohira, with one of the best half day hikes you'll ever take in your life.
https://www.youtube.com/watch?v=ley9rVIewoQPlus if you wanted to allocate a couple of days for Kyushu, they would not be wasted time at all. Along the way in Yamaguchi is Akiyoshido, which is a 300 million year old limestone cave 1km deep with an underground river and giant caverns. It's truly a marvel and one of Japan's best yet largely unknown sights.
https://www.youtube.com/watch?v=DqIKC9S8TFQFukuoka has a number of good places to see if you just look. Dazaifu is OK, but even better is the Nanzoin Temple about 20 minutes by train from Hakata Station. A lot of other temples may turn into a blur by the end of your trip, but
not this place.
https://www.youtube.com/watch?v=lT4VQFmHFyMInside the city are 3 beautiful tradition gardens at Rakusuien, Yusentei and Ohori Park, plus the views from Fukuoka Tower and the Minami Koen Tenbodai are excellent. Nokonoshima has a nice relaxing hike and island park garden that's great too.
http://www.thejapanfaq.com/fukpic.htmlhttps://www.youtube.com/playlist?list=PL7LcPcmoSg-j4RLa6iC7ayvBAbOn8-0...Himeji would be an excellent choice as well to see Japan's best castle, and don't miss the Kokoen Garden next to it. I'd also say you can largely skip Osaka - but take a day and overnight and go see Koyasan. It is one of the deeply spiritual sides of Japan and once you will find unforgettable.